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THE 


PRIMITIVE   DOUBLE    MINIMAL   SURFACE  OF 


THE  SEVENTH   CLASS 


AND 


ITS  CONJUGATE. 


Submitted  in  partial  fulfillment  of  the  requirements  for  the  degree  of  Doctor  of 
Philosophy,  in  the  Faculty  of  Pure  Science,  Columbia  University, 


BY 


GRACE    ANDREWS, 


^■:^, 


/; 


NEW   YORK. 


1901 


THE 

PRIMITIVE   DOUBLE    MINIMAL   SURFACE   OF 

THE  SEVENTH    CLASS 

AND 

ITS  CONJUGATE. 


Submitted  in  partial  fulfillment  of  the  requirements  for  the  degree  of  Doctor  of 
Philosophy,  in  the  Faculty  of  Pure  Science,  Columbia  University, 


BY 


GRACH    ANDREWS 


^    OF  THE 

UWfVERSlT\' 

or 


NEW    YORK. 

The  Evening  Post  Job  Printing  House,  156  Fulton  Street. 

(evening  post  building.) 

I  9  o  I  . 


A- 


^^ 


^vlv. 


THE  PRIMITIVE  DOUBLE   MINIMAL  SURFACE  OF 
THE  SEVENTH  CLASS  AND  ITS  CONJUGATE. 


In  a  paper  entitled  "On  Certain  Algebraic  Double  Minimal 
Surfaces,"  the  author,  Dr.  James  Maclay,  discusses  the  minimal 
surfaces  determined  by  the  following  conditions  :  (i)  The  gen- 
erating minimal  curves,  conveniently  designated  by  (a)  and 
(/^)  shall  coincide,  that  is,  the  curve  (a)  shall  be  its  own  con- 
jugate ;  (2)  to  one  point  of  the  infinitely  distant  circle  shall 
correspond  one  tangent  to  the  curve ;  (3)  the  curve  shall  have 
only  two  infinitely  distant  points. 

These  points  may,  without  loss  of  generality,  be  taken  to 
correspond  to  7i  =  0  and  u  =  co.  When  this  is  done  and 
the  functions  F{!()  and  /(?/),  [/'"{?()  =-  F{u)l  that  define  the 
surface  in  connection  with  the  equations  of  Weierstrass  are 
determined  so  as  to  satisfy  the  prescribed  conditions,  they  are 
found  to  take  the  forms 

(I)  A  (^0 


A- 

--2- 
1 

1 

+ 

^0 

+ 

^1 

IC 



cj 

u' 

^M 

-+    (- 

-i)^  A' 

2 

^M 

= 

— 

-/< 

(/' 

+ 

0(/' 

+ 

2) 

(2)      F,{u)=^:£6^ 
1 

On  assuming  the  a  to  be  included  in  the  A 's  and  putting 


equation  (2)  becomes 

(4)  F,  {u)  =  1  3^. 


1 


The  elements  ^^{u),  (/<  =  i,  2  ....  k)  composing  Fi,{ti) 
themselves  define  double  surfaces.  To  a  given  value  of  ju 
correspond  an  infinity  of  surfaces  arising  from  the  variation 
of  the  arbitrary  constant  A^.  These,  however,  are  all  similar 
since  s^  can  be  thrown  into  the  form 

(5)  ^,  =  c,  '  +  ^''l^^f'  "  '.  (^.  real). 

On  the  other  hand,  except  when  k  ^  1  and  F^  (u)  reduces  to 
3t  the  surfaces  defined  by  F^  (u)  are  not  all  similar.  Since 
the  class  of  F^.  and  s^.  is  the  same,  being  expressed  through 
the  equation  C  =  2;/  +  3,  F^  may  appropriately  be  called  the 
general  ;  5^,  the  primitive  surface  of  the  class  {2k  +  3). 


210501 


The  present  paper  is  a  study  of  the  primitive  double  sur- 
face and  its  conjugate  corresponding  to  //  =  2.* 

I. — The  Primitive  Double  Surface,  /x  =  2. 

I.   T/te  Equations  of  the  Surface. — The   function  defining 
the  surface  is 

(6)  ^•^  =  -^'     (^2=1) 

or, 

(7)  &2  =  ''-—^A&2"    =^2). 

24 

When  f)^  is  substituted  for  f  {u)  in  the  second  form  of  the 
equations  of  Weierstrass,  namely 

x  =  ^{i-u'')f'{tc)  +  uf{:ii)-f{tc)  +  -^  (i-^)  A"i^') 

+  vf,'  {V)  -/i  (V) 

+  ivf,'  {v)-if^{v) 
z  =  lif"  iu)  -/'  {ii)  +  vf{  {v)  -//  (7/) 

the  following  equations  for  the  surface  are  obtained : 

_  I  r     u^  ^  u-^      u^  +  u.-^-\       I  r_  7/*  +  7/  -^ 
^  ~TL  4         "*"         2        J2L  4 


-f 


4 

V  -\-  V   - 
2 


]  =  ^  +   ^1 


tl  4  2  —  2-1  '1-4 

^^— +^ — J-tL— T^ 


1?  —  u      ^    ,    7/'*  V      ^ 


+ =N+N,. 

i  3 

The  paper  already  cited  determines  the  plane  lines  of 
curvature  and  the  straight  lines  of  the  general  surface  F^  (//), 
the  properties  of  the  minimal  curve  and  the  infinitely  distant 
elements  of  the  primitive  surface  ;  the  nature  of  the  geodesic 
in  the  .i-j-plane  by  which  the  primitive  surfaces  /'  even  are 
characterized  and  the  remaining  curves  of  intersection  in  the 
same  plane. 

2.  Plane  Lines  of  Curvature  and  Straight  Lines  of  ike 
Surface. — When  a  surface  is  represented  upon  the  unit  sphere 

*  The  double  surface  corresponding  to  /n  =  i  has  been  treated  by  Schilling  in  the  Inaugural 
Dissertation,  "  Die  Minimalflachen  fiinfter  Klasse." 


through  parallel  normals,  plane  lines  of  curvature  pass  into 
small  circles  with  planes  parallel  to  the  lines  of  curvature  and 
right  line  asymptotes  into  great  circles  with  planes  perpen- 
dicular to  the  right  lines.  It  is  therefore  possible  by  forming 
the  equations  of  the  stereographic  projections  of  these  circles 
on  the  plane  Z  =  o  and  combining  them  with  the  differential 
equations  of  the  plane  lines  of  curvature  and  straight  lines  to 
determine  the  latter. 

When  this  is  done,  it  is  found  that  no  non-meridian  circles 
represent  curves  of  the  desired  type,  except  the  pole  iiv  =  ^, 
which  as  it  involves  u  —  o  corresponds  to  a  line  at  infinity, 
and  the  equatorial  circle  of  the  sphere  iiv  —  \  =  o.  The 
relation  uv  —  i  =  ^  determines  upon  the  surfaces  }x  even,  a 
line  of  curvature  in  the  -i'_j'-plane  and  upon  the  surfaces  /i  odd, 
the  ^-axis. 

It  is  found  that  meridian  circles  cannot  represent  curves 
of  the  desired  types  unless  the  arbitrary  constants  y^^  involved 
in  Fk  {u)  fulfill  certain  conditions.  When  every  A^x.  but  one 
is  zero,  /\.  reduces  to  3^.  In  this  case,  the  surface  possesses 
(/i  4-  i)  lines  of  curvature  lying  in  planes  that  pass  through 
the  .c-axis,  and  (/<  +  i)  straight  lines  lying  in  the  A'j/-plane 
and  passing  through  the  origin.  When  /<  is  even,  the  planes, 
among  which  is  found  the  .r;:r-plane,  pass  through  the  lines, 
which  include  the  .v-axis. 

As  all  the  plane  lines  of  curvature  on  the  surface  are 
represented  on  the  sphere  in  great  circles,  they  are  geodesies, 
and  by  a  theorem  due  to  Schwarz  determine  planes  of  sym- 
metry. By  a  second  theorem,  the  straight  lines  are  axes  of 
symmetry. 

It  follows  from  these  results  that  the  surface  /^  =  2  has 
four  planes  of  symmetry,  including  the  xy  and  .r.c-planes  and 
three  axes  of  symmetry,  including  the  x-axis. 

3.  The  Minimal  Curve. — The  order,  class  and  rank  of  the 
minimal  curve  of  the  primitive  surfaces  are  found  to  be 
2}x  +  4,  2/i  -f  2,  2yM  -\-  4,  respectively. 

The  stationary  points  of  the  curve  are  furnished  by  the 
values  of  11  satisfying. 

(10)  '^^{7i)=o;ox,u^^'^-^  =  {—x)^^\ 

When  /<  is  even,  a  pair  of  conjugate  points  corresponding  to  u 
and  —  II  are  found  to  be  symmetrically  disposed  with  respect 
to  the  xy-plane.  The  middle  point  of  the  line  joining  a  pair 
lies  on  the  surface. 

The  plane  of  osculation  at  the  infinitely  distant  points  of 


6 

the  curve  falls  into  the  plane  at  infinity  and  the  tangent  line 
becomes  the  tangent  to  the  imaginary  circle. 

When  /^  =  2,  the  order,  class  and  rank  are  8,  6  and  8 
respectively,  and  equation  (lo)  takes  the  form 

(11)  U^  =  —\. 

4.  TJie  Infinitely  Distant  Elements. — These,  as  is  known,  lie 
on  straight  lines  obtained  by  joining  the  infinitely  distant 
points  of  one  minimal  curve  to  the  infinitely  distant  points  of 
the  other.     They  arise  therefore  from  the  four  combinations 

{\)    II  =0,V  =  0  \     (2)    ?/  =  CO  ;     V  =   ^\     {^)    U  =  0,  V  =  CO  ;     (4) 

?^  =  00 ,  V  =  0.     When  these  values  are  introduced  into   the 
equations  of  the  primitive  surfaces 

X  +  iy  =  ^- ^ , j-— -  +  ^^ '- 

,       ,  .  —  «  -  '^  -  2    ,     (—  l)^  «AX         (_  l)'^  +  '  Zy'^  +  ''        7/  -  '^ 

(12)  x  —  ty^ —J— — H j H  


/<  +  2  /f  /'  +  2.  ^l 

^  ^  /i  +    I.  /'  +    I. 

it  is  found  that  combinations  (i)  and  (2)  give  rise  to  the  line 

(13)  Z  =  0.       T  =  0. 

and  (3)  and  (4)  to  the  lines 

(14)  X  -\-  iy  =  0.      z  =  0. 

(15)  X  —  iy  ^  0.     T  =^  0. 

respectively,  r  =  ^  is  the  equation  of  the  plane  at  infinity. 
The  lines  are  not  simple  lines  of  the  surface.  This  may  be 
shown  in  the  case  of  the  first  line  as  follows  :  assume  ?/  =  00  , 
?;  =  00  ,  and  regard  only  the  terms  of  the  highest  degree  in 
equations  (12).     We  have  then 

/ i)^  +  1  ^M-  +  2 


(16) 

.r  + 

z> 

+  1  ^^,x  +  2 
+  2.        ■ 

from  which 

we 

derive 

(17) 

X 
X 

+  zy  _  / 

—  iy        ^ 

X  —  z  y 


H  +  2. 


2 


To  every  point   of   the    straight   line   correspond  (yu  +  2) 
values  of  the  ratio  — ,  when  ?^  =  oo  ,  t--  =  oo  . 

V 

But  now  assuming  u  =  o,  v  =  o,  and  regarding  only  the 
terms  of  lowest  degree,  we  have 

^-»x-2  _  Z^-^-2 

(18)  x  +  iy  =  —     ,,    ■    ^    ,   x  —  ty  = 


/<  +  2  /<  +  2 


and  therefore 

(19)  -iL±i>="''^"^=.(^y"' 

^  ^'  x  —  iyu-*^-^        \v  ) 

To  every  point  of  the  straight  line  correspond  (/<  +  2) 
values  of  the  ratio  — ,  when  11  =  0,  v  =  0.      Apparently  the 

V 

line  is  of  multiplicity  2(/'  +  2).  But  for  a  given  point,  to 
every  pair  of  values  Ji,  v  satisfying  (16)  and  (17)  correspond  a 
pair  —  T'  ~',  —  u  ~^  satisfying  (18)  and  (19).  It  is  character- 
istic of  double  surfaces  that  every  point  corresponds  to  two 
pairs  of  values  of  the  parameters,  namely,  u,  v  and  —  v" , 
—  jr^  therefore  the  multiplicity  of  the  line  under  considera- 
tion is  (/<  -(-  2). 

The  two  imaginary  lines  (14)  and  (15)  are  also  of  multi- 
plicity {i-i  +  2). 

5.  The  Geodesic  in  the  xy-Plane. — By  employing  the 
equations  of  Schwarz  defining  a  minimal  surface  in  terms  of 
a  curve  through  which  the  surface  passes,  the  following 
relation  is  obtained  between  p  the  radius  of  curvature  of  the 
geodesic  and  the  parameter  u 

(20)  p  =  li-  S(ii  {u). 

It  ^b 

Through  the  substitutions  ii  =  e'^  and  qi  = -^,   (20) 

may  be  changed  into 

(21)  P  =  4  sin Tp. 

In  this  form  p  is  recognized  as  the  radius  of  curvature  of  a 
hypocycloid  generated  by  means  of  a  fixed  and  a  rolling  circle 
with  radii  equal  respectively  to 

1.1  {/ii  +  2)  /'  +  2 

ip  is  the  angle  between  the  line  of  centers  of  the  two  circles 
and  the  .f-axis. 

From  the  relation  between  R  and  r,  it  is  seen  that  the 
curve  has  (/<  +  i)  cusps,  one  of  which  lies  on  the  ,t'-axis. 
Since  a  cusp  occurs  wherever  p  =  o,  a.  comparison  of  (20)  and 
(10)  shows  that  the  points  in  the  ,t-j/-plane  furnished  by  pairs 
of  conjugate  stationary  points  of  the  minimal  curve  are  the 
cusps  of  the  hypocycloid. 

6.  The  Curves  of  Intersection  in  the  xy-Flane. — It  is  of  ad- 
vantage to  determine  these  for  the  surface  /'  =  2  by  a  special 
discussion  and  to  reserve  until  later  a  statement  of  the  more 
general  results  reached  for  the  surfaces  /<  even. 


When  for  u  and  v  are  substituted  A  f  '''  and  A  f  '  *  equation 
(9)  takes  the  form. 

;r  =  —  i  (A^  +  A-^)  cos  40  +  i  (A"  +  ;\-")  cos  20 

(22)  J  =  —  T  (A*  +  A--*)  sin  40  —  1  (A"  +  A-^)  sin  20 

2-  =  I  (A^  —  A~')  cos  30. 

The  substitution  p  =  A  —  A  -^  obviates  the  difficulties  that 
arise  from  the  fact  that  each  point  of  a  double  surface  corre- 
sponds to  one  value  of  the  parameter  0,  combined  with  two 
values  of  the  parameter  A^  namely  A  and  —  A  ~\  By  this  sub- 
stitution equations  (22)  become 

.a:  :=  —  i  (/3*  -f  4P  +  2)  cos  40  +  i  (p"  +  2)  cos  20 

(23)  y  =  —  \  {P*  +  4P^  +  2)  sin  40  —  ^{p-^  2)  sin  20 

■s-  =  t  P  (P"  +  3)  cos  30. 

The  curves  of  intersection  in  the  Ar;'-plane  are  determined 
by  the  conditions 

{A)p  =  o,      {B)p^+2  =  o,     (C)  cos  30  =  ^.- 
Condition  (A)  leads  to  the  equations 

X  =  —  (cos  1p  -\-  ^  cos  21^') 

(24) 

y  =^  —  (sin  ip  —  i  sin  2ip). 

where  ip  ='  n  —  20.  These  define  a  hypocycloid  generated 
by  a  fixed  and  a  rolling  circle  of  radii  3/2  and  1/2  respectively. 
The  radius  of  curvature  p   satisfies  the  equation 

(25)  p'  =  4  sin  3/2  ^'  =  4  cos  30. 

The  curve  therefore  has  three  cusps,  one  of  which  is  on 
the  or-axis  at  x  =  —  3/2.  The  corresponding  vertex  is  at 
X  =  1/2. 

Since  A  =  i  and  7(v  —  i  =  o,  when  p  =  o,  the  curve  is  the 
geodesic  in  the  ,i'jj/-plane  already  mentioned. 

Its  degree  may  be  determined  by  putting  v  =  11  ~'  in  equa- 
tions (9)  and  substituting  the  resulting  values  of  :r  and  j  in  the 
equations  of  an  arbitrary  line.  This  gives  an  equation  of  the 
fourth  degree  in  li^.  The  degree  of  the  curve  is  four.  The 
multiplicity  is  apparently  two,  but  is  really  one,  since  the  two 
pairs  of  values  of  the  parameters  to  which  each  point  of  a 
double  surface  corresponds,  namely  «,  v  and  —  v~^^  —  /<  ~^  be- 
come for  points  on  the  curve  in  question  u^  v  and  —  ?/,  —  v. 


Condition  {B)  leads  to  the  equations 

X  =  2"   (cos   l})  -\-  ^  cos  2^/') 

(26) 

y  =\  (sin  ip  —  i  sin  2^'). 

These  evidently  define  a  hypocycloid  derived  from  the 
preceding  by  means  of  the  proportional  factor  —  1/2.  The 
curve  has  a  cusp  at  x  =  3/4,  y  =  o  and  a  vertex  at  ,i'  =  —  1/4, 
J/  =  o.  It  is  double  since  each  point  corresponds  to  two  vahies 
of  p  and  is  isolated  since  these  values  are  imaginary;  for 
an  infinitesimal  change  in  p  makes  the  co-ordinates  of  the 
surface  imaginary. 

Condition  [C)  implies  the  relations 

sin  6(p  ^  sin  40  cos  20  +  cos  40  sin  20  =:  0 

(27)  sin  4(p   cos  41?? 

sin  2(p  cos  2cp 

The  equations  of  the  section  therefore  become 


x       _  y 


(28)  cos  20  sin  itp 


i  (p^  +  6  fy  +  6] 


^  o. 


These  are  the  equations  of  three  straight  lines  of  multi- 
plicity 4,  namely  : 

y  =  0,  7  =  ±  V  3  .r. 
Regarded  as  an  equation  in  p",  (28)  is  satisfied  by 


(-9)  p'  -  -  3  ±  ^  3  + 


4.r 


cos  20 


Two  of  the  four  values  of  p  are  real  if  C"  ^h      Other- 
cos  2Cp   >  "^^ 

wise  all  are  imaginary.  Comparison  with  equations  (24)  and 
(25)  shows  that  the  projections  of  the  cusps  of  the  geodesic 
hypocycloid  upon  the  Ar-axis  are  given  by 

-^  =^  3/2  cos  2  0,  when  cos  ^(p  ^-.  0. 

The  straight  lines  therefore  proceed  outward  from  the  cusps 
of  the  hypocycloid  to  infinity  in  connection  with  real  parts  of 
the  surface.     Two  real  nappes  pass  through  each. 

7.  Order  of  tJic  Surface. — On  adding  together  the  degrees 
of  the  vari'ous  curves  of  intersection  in  the  .vj-plane,  regard 
being  paid  to  the  multiplicity  of  each,  the  order  of  the  surface 
is  found  to  be  28.  This  accords  with  the  result  obtained  from 
the  formula 

C  =  f  w  (w  —  i), 

when  m  is  the  order  of  the  minimal  curve. 


10 

There  are  in  the  a:_>'-plane 

3  finite  straight  lines  of  multiplicity  4 12 

I  straight  line  at  infinity  of  multiplicity  4 4 

I  hypocycloid  of  multiplicity  i,  degree  4 4 

I             "           "           "           2,       "       4 8 

28 

8.  Curves  of  Intersection  in  the  xz- Plane. — These  are 
determined  by  the  conditions 

{A)  cos  cp  ^  o,    {B)  sin  q)  ^=  o,    (Q  cos  2^  = A, 

(A)  is  the  condition  for  the  a;-axis. 

(B)  leads  to  a  curve  of  the  fourth  degree  with  the 
equations 

X  =  —  i  (^<  +  2  p^  —  2). 
(30) 

y  =  ip{p-  +  3). 

The  values  of  the  direction  cosines  of  the  normal  to  the  sur- 
face at  a  point  n,  v  are 

(31)  x  =  ^tA^,     F  =  /^^^.    z  =  i^^. 

uv  -\-  \  uv  -\-  \  uv  -}-  I 

Since  n  =  v  when  sin^  =  o,  the  normals  to  the  surface 
along  the  curve  defined  by  (30)  fall  into  the  a-^-plane.  The 
curve  is  therefore  a  geodesic. 

The  slope  of  the  tangent  at  any  point  is  derived  from 

cix  =  —  p  {p-  +  i)  itp,  dz  ^  2  [p-  -}-  i)  dp. 
dz 

The  substitution  of  —  p  for  p  shows  that  the  curve  is 
symmetric  with  respect  to  the  .:v-axis  which  it  cuts  orthogon- 
ally at  .V  =  1/2  and  in  an  isolated  double  point  at  jv:  =  —  1/4. 
In  addition  to  this  singularity,  it  has  two  imaginary  cusps 
corresponding  to  the  roots  of  p-  +  i  =  o.  Since  it  is  unicursal 
and  of  degree  4,  it  can  have  no  other  double  points. 

It  cuts  the  .c-axis  in  four  points,  two  real  at  .;•  =  ±  2.128 
and  two  imaginary.  It  proceeds  to  infinity  without  maxima, 
minima  or  points  of  inflection  and  becomes  parallel  at 
infinity  to  the  .r-axis.  It  is  of  the  type  of  the  ordinary 
parabola. 


11 


Condition  (C)  gives  rise  to  a  double  curve  of  the  tenth 
degree  with  the  equations 

_^{p'  +  6  ,/  +  6)  {p'  +  2  p2  -2) 
4 


X 


(32) 


2   := 


V 


P*  4-  4  P^  +  2. 


The  curve  cuts  the  :i:-axis  at  the  origin,  an  isolated  point, 
and  at  ^  =  —  3/2  and  .v  =  3/4,  cusps  of  the  hypocycloids  in  the 
ATj'-plane. 

Since  to  one  value  of  x  correspond  four  values  of  p\  the 
curve  has  eight  branches  arranged  in  pairs  symmetric  to  the 
;t;-axis.  On  throwing  the  equation  for  x  into  the  form  of  a 
biquadratic  in  r,  (r  =  p^)  and  forming  the  discriminant 

J  =  64/9  (18  .r'  +  82  X*  +  216  x^  +  306  -t-  +  270  -r  -f  81) 

it  is  found  that  the  only  real  value  of  x  for  which  A  is  zero 
is  one  that  lies  between — .5  and — .6  and  may  be  conveniently 
designated  by  a,,.  For  x  <  a,,,  two  values  of  r  are  imaginary; 
for  0:  >  «o,  all  are  real.  For  —  3/2  <  x  <  00,  one  of  the  real 
roots  is  positive.  This  corresponds  to  the  non-isolated  part 
of  the  curve. 

When  with  these  results  are  combined  the  facts  presented 
in  the  following  table,  showing  the  comparative  values  of  ^ 
a;  and  z 


—  4-7 
+       o 
Imaginary. 


3-4 


Real. 


—3 
-h 
o 


2.7  . .  —  1.2 
+  o    -        o      + 

Imaginary 


-.58 


Real. 


•73 

o 

Real. 


the  conclusion  is  reached  that  the  curve,  so  far  as  it  is  real, 
consists  of  two  detached  parts;  the  first  wholly  isolated,  asso- 
ciated with  values  of  r  ranging  from  —  3.4  to  —  3;  the  second 
partly  isolated,  associated  with  values  ranging  from  —  .58 
to  00  . 

Further  light  is  shed  upon  the  course  of  the  curve  through 
the  values  of  dx  and  dz. 


dx  = 


^  r'^  _|_  10  r*  -f  36  r-^  +  56  r^  +  44  r  +  24 


dr 


d2  = 


v/" 


/         ^"+3         V/^     r^+  ior*  +  36r^  +  56r-+44r-f  24 
V  r-^  -1-  4r  +  2   /  (r2  +  4r  -f  2)^  ' 


1/2 


+  4r  + 

dx       ^  ^    V  r2  -I-  4r  +  2  / 
Since  the  equation 

r^  _}-  10  r^  -f-  36  r'^  +  56  r^  -f-  44  2"  +  24  =  o 


dr 


12 

has  but  one  real  root  and  that  lies  between  o  and  —  3;  and 
the  equation 

r^  +  4r+2=o 

has  the  roots  —  3.4  and  — .58,  the  common  factors  of  dx  and 
dz  determine  no  cusps  upon  the  real  branches  of  the  curve. 

The  wholly  isolated  part  of  the  curve  resembles  a  semi- 
cubic  parabola  with  a  cusp  on  the  ;r-axis  at.T=  3/4,  from  which 
point  the  curve  extends  in  the  positive  direction  to  infinity. 
The  second  part  consists  of  two  branches  parallel  to  the  xr-axis 
at  2;  =  00,  5-  =  rh  00  when  they  take  their  rise,  cutting  each 
other  at  an  angle  of  120°  on  the  .T-axis  at  x  =  —  3/2,  proceed- 
ing from  this  point  in  connection  with  real  parts  of  the  sur- 
face, intersecting  the  geodesic  in  the  two  points  ±  2.128  on 
the  ^-axis  and  becoming  parallel  to  the  .;i:-axis  at  infinity. 

To  recapitulate,  the  curves  in  the  jr^-plane  are: 

I  straight  line  (the  .r-axis)  of  multiplicity  4 4 

I  geodesic  of  degree  4,  and  multiplicity  i 4 

I  curve  of  degree  10,  and  multiplicity  2 20 

28 

9.  Symmetry  of  the  Surface. — It  is  unnecessary  to  make  a 
special  study  of  the  curves  of  intersection  in  the  planes 
_y  =  ±  v^3  X.  The  sections  of  a  surface  by  a  set  of  planes  of 
symmetry  that  pass  through  a  straight  line,  as  the  ;?-axis,  and 
are  so  related  that  the  angles  between  consecutive  planes  are 
equal,  cannot  be  of  more  than  two  types;  for  any  one  of  the 

planes  (i),  (3),  (5) is  symmetric  to  some  other  of  the  same 

group  with  respect  to  some  one  of  the  planes  (2),  (4),  (6) 

and  vice  versa.  If  «,  the  number  of  planes,  is  odd,  the  w'^ 
plane  belongs  to  the  same  group  as  the  first,  but  at  the  same 
time  is  symmetric  to  the  second  plane  with  respect  to  the 
first,  so  that  the  two  types  of  section  furnished  by  consecutive 
planes  can  differ  only  in  the  position  of  their  component  curves 
with  respect  to  the  origin. 

In  the  present  case  since  the  curves  in  the  .I'^-plane  are 
symmetric  to  the  a;-axis,  the  two  types  of  section  made  by  the 
three  vertical  planes  differ  only  in  the  relation  of  their  curves 
to  the  ^•-axis.  The  geodesies  and  double  curves  in  all  three 
planes  pass  through  the  same  two  points  on  the  ^-axis.  These 
are  singular  points  for  the  surface  of  multiplicity  three. 

The  determination  of  the  planes  and  axes  of  symmetry  of 
a  minimal  surface  by  means  of  its  plane  geodesies  and  straight 
lines  is  not  necessarily  exhaustive  since  the  theorems  con- 
verse to  those  of  Schwarz  are  not  true.     But  that  in  the  pres- 


13 

ent  instance  it  is  complete  is  readily  seen.  If  the  double  sur- 
face 1-1  —  2  should  possess  an  additional  plane  of  symmetry, 
the  latter  must  bisect  the  angle  between  two  of  the  planes 
j/  =  <?,  j/  =  rb^3  X  since  in  any  other  position  it  would  re- 
quire the  existence  of  a  second  set  of  straight  lines  lying  on 
the  surface.  The  existence  of  one  such  plane  would  imply 
the  existence  of  three  bisecting  the  angles  between  the  planes 
of  symmetry  already  found,  and  presenting  a  third  type  of 
section.  As  no  such  type  can  exist,  neither  can  the  assumed 
planes. 

There  can  be  no  additional  axes  of  symmetry ;  for  they  could 
be  no  other  than  a  set  of  three  bisecting  the  angles  between 
the  straight  lines  of  the  surface,  and  in  this  position  since  the 
surface  is  symmetrical  to  the  a;j/-plane,  they  would  with  the 
^-axis  determine  three  additional  planes  of  symmetry. 

It  is  possible  to  place  some  limitation  on  the  nature  of  the 
curves  of  intersection  of  an  algebraic  surface  and  a  plane  of 
symmetry.  Such  curves,  if  neither  imaginary  nor  multiple, 
must  be  geodesies.  If  F  (,r,  j/,  :;)  =  o  represents  an  algebraic 
surface  referred  to  a  plane  of  symmetry  as  the  a'j'-plane, 
F  (.?•,  J/,  ^)  contains  only  even  powers  of  ;;•  and  dFjds  contains 
odd  powers  in  every  term.  The  direction  cosines  of  the  normal 
at  (:r,  J,  z)  are  : 

d  F  dF 

9  X  d  y 


\  m '+ (1-^) '+ m  "■  ^  (H)  --  (Q '+ a)  ■ 


d  F 

dz 


mr-i^y-m) 


dF 
Along  a  curve  of  section  in  the  jrj/-plane,  ^^  =  o.      Un- 


less   at   the   same   time        / \    4-  / \    _l    / \     =  o 

\1{9  x)     ^  \9y)    +    K9  ::) 

that  is,  unless  the  curve  is  multiple,  the  normal  to  the  sur- 
face falls  into  the  arj/-plane  and  coincides  with  the  normal  to 
the  curve.     The  latter  is  therefore  a  geodesic. 

10.  Curves  Corresponding  to  the  Parallels  and  Meridians  on 
the  Sphere. — These  constitute  two  systems  cutting  each  other 
orthogonally  and  obtained  respectively  by  means  of  the 
relations 

(33)  «^  ^}?  ^c, 


14 

and 

(34)  V  ^  ft  u  or  0  =  k. 

The  A  Curves. — The  A  curves,  as  they  may  be  called,  are  of 
the  eighth  degree,  with  the  exception  of  two  curves  of  the 
fourth  degree,  namely,  the  geodesic  hypocycloid,  uv  =  i,  and 
the  real  line  at  infinity,  uv  =  0. 

Since  Af  '^  =  —  As '  (-r  +  -l>),  a£  ~  *  *  =  —  Ae  ~ '  <"  +  ^),  and  on  a 

double  surface  Af '*    Af  ~  '*  and  —si(^+4>),  —   e  ~  ^  ('^  +  '^)   fur- 

A  A 

nish  the  same  point,  all  of  the  curves  of  the  system  are 
obtained  by  assigning  to  A  values  ranging  from  i  to  co  .  A 
being  fixed,  the  substitutions  cp,  n — qj,  7t  ^  (p  show  the 
curves  to  be  symmetric  with  respect  to  the  xy-  and  xs- 
planes  and  to  the  .i'-axis,  and  therefore  to  all  the  planes  and 
axes  of  symmetry. 

The  curves  are  closed.  With  the  exception  of  the 
geodesic  hypocycloid,  each  curve  cuts  the  .i^^-plane  in  three 
double  points,  one  on  each  of  the  straight  lines  of  the  surface 
and  cuts  each  vertical  plane  of  symmetry  in  a  double  point 
on  the  straight  line,  two  ordinary  points  on  the  geodesic  and 
two  double  points  on  the  double  curve.  One  curve  of  the 
system  cuts  the  ^-axis  in  the  two  singular  points.  No  two  of 
the  curves  intersect. 

The  cp  Curves. — These  are  of  the  fourth  degree.  They 
include  the  vertical  geodesies  and  the  straight  lines  of  the 
surface.  All  the  curves  of  the  system  are  obtained  by  assign- 
ing to  cp  values  ranging  from  0  to  ;r,  and  all  the  points  of  a 
curve,  by  permitting  \  to  vary  from  0  to  ^  . 

Each  curve  is  symmetric  with  respect  to  the  ;v^-plane  and 
to  cp  and  n  —  cp  correspond  curves  symmetrically  situated 
with  respect  to  the  rr^-plane  and  intersecting  upon  the  double 
curve  in  that  plane  if  45°  ^  9?  ^  90°. 

From  the  direction  cosines  of  the  tangent 

—  (A'^  —  i)  cos  \(p  +  A"'  (A*  —  i)  COS  ^cp 
P  1/2  '-         ' 

—  (A'"*  —  i)  sin  4(p  —  A-  (A^  —  i)  sin  2<p         2  (A"  +  i)  cos  3(p 

P  1/2  '  P  z/2 

where 

P  1/2  = 
^(A8_i)24-(A«— A2)^— 2(A«— OCA"— A-)cos6<p  +  4A^  (A^  +  i)^  cos^3<p 

it  is  evident  that  the  curves  cut  the  geodesic  hypocycloid 
orthogonally  except  at  the  cusps  and  become  parallel  to  the 
arK-plane  at  infinity. 


15 

II.  Model  of  the  Surface. — By  making  a  framework  of 
cardboard  representing  the  non-isolated  curves  of  inter- 
section in  the  planes  of  symmetry,  and  using-  this  to  support 
a  network  of  wires  representing  certain  q)  and  A  curves,  1 
constructed  a  skeleton  model  of  the  surface.  The  following 
values,  multiplied  throughout  by  two,  constitute  the  data 
for  the  construction. 


Geodesic 

Geodesic  in 

Double  Curve  in 

Hypocycloid. 

X2- 

Plane 

xs-Plane. 

i> 

X 

y 

P 

X 

^ 

r[=ff) 

X 

z 

oo 

—1.50 

0 

0 

•50 

0 

0 

—  1-5 

0 

20° 

— I 

32 

—  .02 

I 

■49 

.20 

.  I 

—  I 

23 

•45 

40° 

— 

«5 

—  -15 

2 

.48 

.40 

.2 

— 

99 

.82 

60° 

— 

2q 

—   -43 

3 

•45 

.62 

•3 

— 

79 

1. 12 

80O 

30 

—  .81 

4 

.41 

.84 

•4 

— 

59 

1^39 

90° 

SO 

— ^i  .00 

5 

■36 

1.08 

■5 

— 

40 

1.63 

iioo 

73 

—  1.26 

6 

.29 

1-34 

.6 

— 

23 

I  85 

120° 

75 

— 1 .30 

7 

.19 

1.63 

•7 

— 

04 

2.06 

130° 

73 

-1.26 

8 

.08 

1.94 

.8 

12 

2.27 

150° 

62 

—  -93 

86 

0 

2. 13 

•9 

29 

2.46 

170° 

?i 

—   -34 

9 

-.07 

2.29 

1 .0 

46 

2.65 

180° 

SO 

0        I 

0 

—  .25 

2.70 

I .  I 

64 

2.83 

I 

686 

-.46 

3-09 

1 .2 

8-- 

3.01 

The  parallels  and  meridians  of  the  sphere  are  conformally 
represented  on  the  plane  Z  =  ohy  concentric  circles  around 
the  origin  and  their  radii.  These  correspond  respectively  to 
A-  =  (7  and  cp  —  k.  They  in  turn  through  the  logarithmic 
function  are  conformally  represented  on  a  second  plane  by 
lines  parallel  to  the  axes  with  equations 

X  =  log  A  and  y  =  q). 


The  A  and  <p  curves  chosen  for  the  construction  of  the 


It 


model  correspond  to.  lines  at  intervals  of  — , 


The  following 


table  presents  the  points  of  intersection  of  the  two  sets  of 
curves.  On  account  of  the  symmetry  of  the  surface  it  is  un- 
necessary to  give  log  A  and  cp  values  greater  than  4^24. 

The  curve  A  =^  1.515  is  introduced  because  it  is  the  A-curve 
that  passes  through  the  multiple  points  on  the  ^--axis. 


16 


Tt 

2Tt 

3'^ 

47ir 

X 

q}  =  0 

^==:77 

^  —  :r. 

<?>  =  — 

=  'P  — 

24 

24 

24 

24 

X 

•5 

•53 

.62 

•71 

•75 

e°=i 

y 

0 

—  -50 

—0.94 

— 1 .21 

—  1.30 

TT 

z 

0 

0 

0 

0 

0 

£    24 

X 

.46 

.50 

.61 

•73 

.80 

=    1. 14 

y 

0 

—  ^55 

— 1 .01 

—  1.30 

— 1^39 

2 

•54 

•49 

•38 

.20 

0 

27r 

£  ^r 

X 

•34 

•38 

•59 

.81 

•97 

=  1.299 

y 

0 

-  .69 

—1.26 

—1. 61 

—  1.68 

z 

1. 16 

1 .06 

•81 

•44 

0 

Sf 

£    24" 

X 

.05 

.18 

•51 

•94 

1.29 

=   1.48 

y 

0 

-  .98 

-1.76 

—2.21 

—2.25 

z 

1 .96 

1. 81 

1-39 

•75 

0 

X 

0 

.14 

.50 

•97 

1^37 

A  =  1.515 

V 

0 

— 1 .04 

—1.86 

—2-33 

—2.37 

ly 

2.13 

1.97 

1. 51 

.81 

0 

1!L 

f  24 

X 

-.46 

—  .24 

•36 

^•13 

1.83 

=  1.686 

y 

0 

—1.44 

-2.58 

— 3^19 

-3.16 

s 

3-07 

2.83 

2.12 

1. 17 

0 

I  am  indebted  to  the  kindness  of  Professor  Hallock  of 
Columbia  University  for  the  model  of  which  two  views  are 
presented  in  Plate  I.  Professor  Hallock  formed  three  shells 
of  paper  upon  a  plaster  impression  of  a  clay  model  that  I  had 
made  of  the  large  concavity  shown  in  Fig.  i.  By  joining 
these  shells  and  supplying  minor  parts  above  and  below  the 
singular  points  on  the  axis  of  2,  he  completed  the  model 
depicted. 


II.  The  Conjugate  Surface  /t  =  2. 

12.  General  Relations  Between  a  Mini)nal  Surface  and  Its 
Conjugate. — By  virtue  of  these,  many  of  the  properties  of  the 
surface  now  to  be  treated  can  be  inferred  at  once  from  those 
of  the  surface  just  discussed. 

As  the  equations  of  the  conjugate  are  derived  from  those 
of  the  original  surface  by  the  substitution  of  iF{ii)^  —  ?'i^(T') 
for  F{ti),  F^  {v)  or  of  if  {u),  —  if  {v)  for  f{u),f  (»),  the  conju- 
gate of  a  double  surface  cannot  itself  be  double  ;  for  if  F[u) 
satisfies  the  functional  relation  for  double  surfaces,  iF{u) 
does  not. 

Corresponding  curves  on  a  surface  and  its  conjugate,  that 
is,  curves  arising  from  the  same  relation   between   u  and  v, 


17 

have  the  same  spherical  image.  Corresponding  elements  are 
orthogonal.  Lines  of  curvature  on  the  one  surface  correspond 
to  asymptotic  lines  on  the  other  and  vice  versa. 

The  origin  is  a  conical  point  on  the  conjugate  of  a  double 
surface.  It  is  also  a  point  of  symmetry.  From  the  charac- 
teristic property  already  noted  that  each  point  corresponds 
to  two  sets  of  values  of  the  parameters,  u.,v  and  —  j)"', —  7/~\  it 
follows  that  the  co-ordinates  of  a  double  surface  may  be 
represented  by  equations 

X  =  (p  (ii)  -f-  cp^  {v)        =  (p{—  7 -')  -f-  ^1  (—  u-^) 

(35)  J=  <^  («)  +  (^i  M         =<^{-  ^~')  +  ^1  (-  «-') 
2  =x  {u)  4-  Xi  (^)  =x{—  ^~')  +  Xi  (—  u~^) 

Evidently  q?  (u)  =  (pi  ( —  n  ~^),  qji  [v)  =  cp  ( —  v  ^^)  and  similar 
relations  connect  c,  Gi  and  j,  X\  so  that  the  coordinates  of 
the  conjugate  surface  become 

X  =  i  [<p  {ii)  —  <Pi  [v]]        =  —  i[<p  (—  7/-^)  —  ^]  (—  //-')] 

(36)  y  =  z['^  («)  -  <?!  (^')]         =-n6  (-  V-')  -  6,   (-  «-')] 
z  =  z[X  («)  -  X,  {^')^  =-nx  (-  v-^)  -  X,  (-  ?^-')] 

These  equations  show  that  to  each  point  of  the  double  sur- 
face correspond  on  the  conjugate  two  points  symmetrically 
situated  with  respect  to  the  origin. 

To  one  curve  upon  a  double  surface  may  correspond  two 
upon  the  conjugate.  The  values  //,  v  and  —  v  ~\  —  21  ~  "^ 
determine  upon  the  unit  sphere  points  (;?■,  J,  z),  ( —  .r,  —  j,  —  z). 
The  complete  spherical  image  of  a  curve  on  the  double  sur- 
face must  therefore  be  symmetric  with  respect  to  the  origin, 
but  it  inay  consist  of  one  curve,  as  for  instance  a  meridian, 
the  two  halves  of  which  are  given  by  the  first  and  second  sets 
of  values  of  the  parameter  respectively,  or  of  two  equal  inde- 
pendent symmetrically  situated  curves,  as,  for  instance,  two 
parallels.  In  the  first  case,  the  spherical  image  will  represent 
a  single  curve  on  the  conjugate  surface ;  in  the  second  case,  two. 

Since  the  two  generating  minimal  curves  of  a  conjugate 
surface  are  derived  from  those  of  the  original  by  a  trans- 
formation that  simply  multiplies  the  distance  of  each  point 
from  the  origin  by  a  constant  factor,  /  in  the  case  of  one  curve, 
—  /in  that  of  the  other,  it  follows  that  finite  points  remain 
finite  ;  infinitely  distant  points  remain  infinitely  distant ;  the 
multiplicity  of  points  is  unchanged  ;  stationary  points  corre- 
spond to  stationary  points ;  coincident  points  of  the  two 
curves  at  infinity  are  unaltered,  as  are  also  the  order,  class 


18 

and  rank  of  the  curves,  the  multiplicity  of  the  imag-inary 
circle  at  infinity  upon  their  tangential  surfaces  and  the  rela- 
tion of  the  plane  of  osculation  and  the  tangent  at  infinitely 
distant  points  to  the  infinitely  distant  plane  and  circle. 

As  it  is  upon  these  properties  that  the  determination  of 
the  infinitely  distant  elements  and  of  the  class  and  order  of  a 
minimal  surface  depends,  it  follows  that  a  surface  and  its 
conjugate  must  have  the  same  infinitely  distant  elements, 
class  and  order  unless  the  original  surface  is  double.  In  this 
case  the  class  and  order  of  the  latter  will  be  and  the  multiplic- 
ity of  the  lines  at  infinity  may  be,  only  one-half  of  the  corre- 
sponding values  for  the  conjugate  surface.  This  is  due  to 
reductions  of  the  type  already  encountered,  resulting  from  the 
double  generation  of  the  double  surface. 

13.  Equations  of  the  Stir  face. — From  these  general  consid- 
erations we  turn  to  the  discussion  of  the  surface  conjugate  to 
the  primitive  double  surface  of  the  seventh  class.  Obviously 
it  is  related  to  the  surface  iF-i  (?/)  as  the  surface  ^2  i")  is 
related  to  that  defined  by  F2  (;/).  It  may  be  called  the 
primitive  conjugate  surface  /<  =  2. 

Its  equations  in  a  form  corresponding  to  (9)  are 

x=  z  [L  —  Zj) 

(37)  y=z{M-Af,) 

^=z(A^— vVi). 

and  in  that  corresponding  to  (22)  are 

X  —  ^  {A.*  —  A~^)  sin  4(p  —  ^  (A"  —  X~-}  sin  2q). 

(38)  .y  =  —  T  (A^  —  A"*)  cos  4<P  —  i  (A-  —  A--)  cos  2cp. 
5r  =  —  t  (A-'  +  A-3)  sin  3<p. 

14.  Pla]ic  Lines  of  Curvature  and  Straight  Lines  of  tJie  Sur- 
face.— To  the  geodesic  in  the  .?j-plane  of  the  double  surface 
must  correspond  an  asymptotic  line  on  the  conjugate.  This 
must  be  a  straight  line  perpendicular  to  the  .a;j-plane  since  its 
spherical  image  is  a  great  circle  parallel  to  that  plane.  Also 
since  every  point  of  the  geodesic  is  represented  on  the  straight 
line  by  two  points  symmetric  with  respect  to  the  origin,  the 
line  must  pass  through  the  origin.     It  is  therefore  the  axis  of  z. 

Similarly  to  the  geodesies  in  the  planes  _;j'  =  o,y^  ±  ■\/~^  x 
correspond  three  straight  lines  in  the  .rj/- plane,  a-=(?,  x^T-  VTy- 
To  the  straight  lines  y  =  o,  z  =  0;  y  =  ±  y^~7  x,  z  =  0  corre- 
spond three  geodesies  in  the  planes  x  =  0,  x  =  ^  y' V  y.     The 


19 

conjugate  surface  is  therefore  characterized  by  four  straight 
lines,  among  which  are  the  y-  and  2;-axes,  and  three  plane 
geodesies,  one  of  which  lies  in  theji^z-plane. 

The  surface  can  have  no  other  straight  lines,  nor  plane 
geodesies,  nor  any  plane  lines  of  curvature  other  than  geo- 
desies since  these  would  imply  the  existence  upon  the  double 
surface  of  plane  geodesies  or  straight  lines  additional  to 
those  already  found  or  of  asymptotic  lines  represented  on  the 
sphere  by  small  circles. 

15,  T/ie  Minimal  Curves. — The  properties  of  these  require 
no  special  discussion,  with  the  exception  of  the  relation  of  con- 
jugate pairs  of  stationary  points  to  the  surface. 

Such  points  are  obtained  by  putting  conjugate  values  of 
It  and  V  derived  from 

«"  =  —  I  and  7/^  =  —  I 

into  the  equations  of  the  minimal  curves  {a.')  and  {ft').  These 
become  for  a  pair  of  values,  ti  and  u~^. ' 

[a')     X  =  2iL,         y  =  ziM,  z  =  2iN, 

(ft')      X  =  —  2/Zi,  /  =  —  2/il/i,  z  =—  2lA\, 

=  —  2lL,  =  —  2zM,         =  2l'N. 

The  conjugate  points  as  was  to  be  anticipated  are  sym- 
metrically disposed  to  the  axis  of  z. 

Equations  (37)  when  a  and  u~^  are  put  for  ?/  and  v  become 


,6 


X  ^=y  =  o 

(39)  ^-^'     "' 

3        'i 

Under  the  condition  ?/'  = —  i,  this  value  of  z  becomes  ±  4/3. 
The  pairs  of  points  under  discussion  are  therefore  connected 
with  two  points  on  the  ;j-axis.  Since  in  the  case  of  the  double 
surface,  such  pairs  are  connected  with  the  cusps  of  the 
geodesic  hypocycloid  from  which  the  straight  lines  of  the 
surface  spring,  we  may  expect  in  the  case  of  the  conjugate 
surface  to  find  them  connected  with  points  on  the  ^-axis  from 
which  spring  the  geodesies.  It  is  to  be  expected  that  the 
relation  of  tangency  between  the  straight  lines  and  the  hypo- 
cycloid  will  be  preserved  between  the  geodesies  and  the  axis 
of  z. 

16.  The  Infinitely  Distant  Elements. — These  lie  upon  the 
same  straight  lines  as  in  the  case  of  the  double  surface,  but 
the  multiplicity  of  the  real  line  in  the  a;j-plane  is  doubled. 

17.  Class  and  Order  of  the  Surface. — These  are  fourteen 
and  fifty-six  respectively. 


20 

1 8.  Curves  of  Intersection  in  the  ,i"j/-plane. — These  are  deter- 
mined by  the  equations 

{A)  A-  4-  I  =  o,    {B)  sin  3^  =  o. 

Under  condition  {A) 

X^  _  X-'  =  A-  —  X--  =  A^  +  A^ 

By  substituting 

A  ^  cos  -^^—2 —  7t  -\- 1  sm  -^—^ —  ^,  [g  an  integer) 

and  reducing,  A^  -|-  A*  may  be  thrown  into  the  form 

zcg  =  2z  sin —  n,  [g  =  o,  i,  2). 

It  —  ip 
Putting  v  =  ,  the  equations  of  the  section  become 

X  =  ic,j  ( —  1/4  sin  2^  4-  1/2  sin^) 

(40) 

y  =  ic,j  (1/4  cos  2^  —  1/2  cos^). 

These  represent  three  imaginary  hypocycloids  with  ver- 
tices on  the  j-axis  as  initial  points.  Of  these  curves  the  origin  is 
one,  appearing  as  a  limiting  case.  It  corresponds  to  the  value 
A2  =  —  T.  For  this  x  =  y  =  z  =  0  irrespective  of  the  value 
of  cp.  The  origin  accordingly  is  a  conical  point  upon  the 
surface.  It  appears  upon  every  qt  curve,  but  in  general  as 
an  isolated  point.  It  is  of  infinite  multiplicity,  but  since  in 
this  case  it  appears  as  a  hypocycloid  it  may  be  regarded  as  of 
the  same  degree  as  the  other  two.  , 

These  may  be  shown  by  means  of  the  relations  uv  =  —  a?, 
nv  —  —  09^  to  be  of  degree  eight  and  multiplicity  one. 

Under  condition  {B). 

sin  4<p  cos  4(p 


sin  2<p  cos  icp 

and  equations  (38)  reduce  to 

,      .  ^^^=_J^= I^(A«^2A«-2A2-I) 

(41)  sin  7.cp         cos  zq>  4A*   ^ 
5  =  o. 

These  are  the  equations  of  three  straight  lines  of  multiplicity 
eight.     From  the  equation 

(42)  A"*  +  2  A''  +  4  { — ^ — ^  A*  _  A-  —  I  =  o 

\  cos  2<p  / 


21  ,,>-^:\^".'-^'  r 

it  is  clear  that  each  line  throughout  itkcpursecis  co^n^ted 
with  two  real  nappes  of  the  surface.  "^"^'^^t.sdLEI''-"" 

The  degree  of  the  surface  may  be  estimated  from  the 
degrees  of  the  curves  of  intersection.  These  are  in  the  vy- 
plane 

3  finite  straight  lines  of  multiplicity  8 24 

1  straight  line  at  infinity  of  multiplicity  8 8 

2  imaginary  hypocycloids  of  multiplicity  i,  degree  8 16 

The  origin 8 

56 
19.    Curves  of  Intersection  in  the  yz- Plane. — These  satisfy 
the  conditions. 

[A]  A^  —  1=0,    [E)  sin  2<p  =  o,    (C)  cos  2<p  = -rv— — ■. 

A    -f-  I 

Condition  (yi)  breaks  up  into  two;  {a)  A^  =  —  i, which  gives 
the  origin  and  {b)  A^  =  i.  The  latter  leads  to  the  equations  of 
the  2-axis. 

.r  ^_y  =  o 

(43)  .    .     , 

^^  =  —  ^  sm  3<p. 

Between  the  points  z  =  ±  4/;^  the  line  is  the  intersection  of 
six  real  nappes  of  the  surface  since  each  of  its  points  corre- 
sponds to  six  real  values  of  cp.  Beyond  the  limits  ±  4/3  it  is 
isolated. 

Condition  (B)  also  breaks  up  into  two  ;  (a)  cp  ^  0,  the  con- 
dition for  the  t'-axis  and  (d)  99  =  tt/i  from  which 


X  =  o 


_y  =.— 


A«_2  A''— 2  A-—  I  (A^_  i)'^(A^4-  I) 


(44)  -"  4A^  4A^ 

A^J   _    .     (A-^+  I)  (A^-A-+  I) 


^=    t 


A.^       ~~    '  A^ 


When  q)  =  7r/2,  ?/  =  —  v,  and  A'  =  o.  The  curve  (44)  is 
therefore  the  geodesic  in  the  jF^-plane.  It  is  of  multiplicity 
one  ana  degree  eight.  It  is  symmetric  with  respect  to  the 
axes.  Aside  from  the  origin  which  appears  as  an  isolated 
point,  it  has  on  the  axis  of  j  two  imaginary  double  points,  and 
on  the  axis  of  z,  two  real  points  ±  4/3. 

On  throwing  the  equation  for  y  into  the  form 

(45)  r^  —  2  r'^  +  4j(/  r-  +  2  r  —  I  =  o,  (r'-  =  A) 

and  forming  the  discriminant 

J=-/(2  7  +  i6/) 


it  is  found  that  J  has  only  one  real  positive  root.  It  follows 
that  to  a  given  value  of  j/ correspond  only  two  real  values  of  z. 
These  are  equal  and  of  opposite  sign.  The  curve  consists  of 
two  detached  parts  symmetrically  disposed  to  the  jF-axis. 
The  value  of  the  derivative  shows  that  each  of  these  resem- 
bles a  semicubical  parabola  with  a  cusp  on  the  2-axis  at  + 
4/3,  or  —  4/3. 

dz=  2  A-*(A"-^—  I)  {1}  +  A^4-l)^/A,  ^  =  — A-^(A''  — i)-(A^  +  A--|-l)^/A 

dz    2  A 

Ify    ^  "~  A^  —  i' 

The  curve  has  four  imaginary  cusps  corresponding  to 
the  roots  of  A^  +  A-  +  i  =  o.  It  becomes  parallel  to  the  axis 
of  J  at  infinity. 

From  condition  {C)  result  the  equations 

.r  =  o 

12        ,\3  n2    1    T^3 


y  = 


(A^-ir(A^  +  ir 


(46)  ^  4  A^  (A^  +  I) 

^  =.  _  ^/T  (A^  +  i)^  (A^  -  A^  +  if' 
^^  A^  (A*  4-  1)3/2 

The  curve  is  double  and  symmetric  with  respect  to  the 
axes.  It  is  of  degree  fourteen.  On  writing  the  equation  for 
y  in  the  form 

^'  -  (3  +  \y)  r~  +  (3  -  4  j)  r  -  I  =  o.  (r  =  A^) 
and  forming  the  discriminant 

A  =  —  64/27/  (4/  — 9) 

it  is  seen  that  the  twelve  values  of  A  corresponding  to  a  given 
value  oiy  are  imaginary,  with  the  exception  of  two  that  are 
equal  and  of  opposite  sign. 

The  curve  resembles  the  geodesic.  It  cuts  the  axes  in  the 
same  points;  consists  of  two  non-intersecting  branches  sym- 
metric to  the  axis  of  >' ;  has  cusps  on  the  ;?-axis  at  ±  4/3  and 
becomes  parallel  to  the  y-axis  at  infinity.  It  differs  from  the 
geodesic  in  the  position  of  its  imaginary  cusps,  determined 
from  the  common  factors  of  dz  and  dy 

(A^-f  i)Ma^-  1)  (A'+4A^  -I-  I) 
A*  (A*  +  1)2 

dz  ,_  A        /A^  — A^-fiV/^ 


A       (r'  —  X'^\\ 
V    2     A  Viri  V      A*  +  I     / 


dj/  1"     ^     A^  —  I  \      A*  + 


23 

There  are  in  the  li-plane  : 

I  straight  line  (s--axis)  of  multiplicity  6 6 

I       "           "     (jz-axis)  "           "             8 8 

I  geodesic  of  degree  8  "           "             i 8 

I  curve        "        "     14"           "             2 28 

Origin ? 

In  this  case  the  origin  enters  through  a  condition  similar 
in  form  to  that  for  the  axis  of  z.  If  it  is  allowable  to  regard 
it  here  as  of  the  same  degree  as  the  2-axis  it  brings  the  sum 
of  the  degrees  of  the  curves  of  intersection  to  the  desired 
figure,  fifty-six. 

20.  Svjuuictry  of  the  Surfaces. — Since  the  curves  in  the  yz- 
plane  are  symmetric  with  respect  to  the  axes,  the  three  planes 
of  the  geodesies  can  yield  but  one  type  of  section.  This  sug- 
gests that  the  surface  possesses  a  second  set  of  planes  of  sym- 
metry. On  substituting  for  Ji  and  v  in  equations  (37),  the 
values  —  //,  —  V  \  —  ■?',  —  ti  ;  —  it  "',  —  i>  ~\  it  is  found  that 
the  surface  is  symmetrical  with  respect  to  all  of  the  co- 
ordinate planes  and  consequently  to  the  axes.  The  ^-^-plane 
and  .r-axis,  then,  belong  to  a  second  set  of  planes  and  axes  of 
symmetry  not  discoverable  throtigh  the  theorems  of  Schwarz. 

21.  Curves  of  Intersection  in  the  xz-Plane. — The  conditions 
to  be  satisfied  are 

As  before,  {A)  gives  the  origin  and  the  2;-axis.  i^B)  leads  to 
a  curve  with  the  equations 


X 


_(A^— i)(A'+A^+i±aV2A«-|-5A^+2)'/''(— 3A-zh-/2A^4-5A*-f2) 
(47)  "^  4-/'^>'tM'^'4-i) 

(A"+i)(2A''+A-4-2+'/2AH-5A^"+2)^^^(A^— A-+I±-v/2A»+5A^+2) 

^~  3A-\A^  +  1)3/2 

For  convenience  of  reference  these  may  be  expressed  in 
the  form 

_    (A^—  i)  A^'^  D 

■^  ~  4  V^A*  (A*  +  I) 

(48) 

^  ^        (A«  +  I)  C'^  D 

3  A^  (A^-+  1)3/2" 
From  the  derivation  of  equations  (47),  it  is  clear  that 

(4Q)  , — =  sin  2<p,  ,.,  .    , c  =  sin  (p. 

^^^^  -/  2    (A*  +  I)  4('^'  +  I) 


24 

The  radical  M  =  -/  2A*  4-  '^K^  +  2  is  real  for  every  real 
value  of  X  and  is  less  in  absolute  value  than  (2A,*  -f  A^  +  2)  or 
than  X -'^  {X^  -{.  X*  -\-  1);  x  and  z  are  therefore  real  for  all  real 
values  of  A.  Since  z  is  imaginary  for  all  imaginary  values  of 
A,  except  those  for  which  it  is  zero  or  infinite,  the  curve  has 
no  isolated  branches. 

On  account  of  the  possible  combinations  of  the  radicals 
involved,  to  a  given  value  of  A  correspond  eight  points. 
These  fall  into  two  groups  of  four  each,  according  as  M  is 
taken  with  the  upper  or  the  lower  signs  in  (47).  The  four 
points  of  a  group  are  symmetrically  situated  in  the  four 
quadrants  ;  rt  A.„  it  i/X„  give  the  same  points.  Apparently 
the  curve  is  of  multiplicity  four,  but  the  consideration  of 
equations  (49)  shows  that  such  is  not  the  case.  When  A  is 
fixed,  the  possible  combinations  of  the  signs  of  A^^  and  6 
indicate  four  related  values  of  ^,  namely,  ^,  tt  —  q),  tt  -{■  q), 
27r  —  cp.  The  change  of  A  to  —  A  gives  rise  to  no  new  values 
of  u  and  v.  The  negative  values  of  a  are  therefore  not  to  be 
counted  in  estimating  the  multiplicity  of  the  curve. 

To  discover  the  nature  of  the  curve,  it  suffices  to  consider 
the  change  in  the  co-ordinates  as  A  ranges  from  i  to  =0 . 
When  X  =  1^  X  —  0  and  z  =  ±.  4/3.  An  examination  of  the 
factors  entering  into  the  numerators  and  denominators  of  x 
and  z  shows  that  they  all  increase  in  absolute  value  with 
A  (A  >  i),  whether  Mhe  taken  with  the  upper  or  the  lower 
signs  throughout.  It  follows  that  the  curve  consists  of  two 
real  branches  in  each  quadrant,  taking  their  rise  on  the  axis 
of  2  at  4-  4/3  or  —  4/3  and  proceeding  to  infinity. 

22.   Curves  Corresponding  to  the  Parallels  anei  Meridians. 

The  A  Curves. — These  are  of  the  eighth  degree,  with  the 
exception  of  the  axis  of  z,  which  is  of  multiplicity  six.  To  a 
curve  A  on  the  double  surface  correspond  two  on  the  conju- 
gate, furnished  by  A  and  i/A.  Each  of  these  curves  is  sym- 
metric with  respect  to  the  xy-  and  j^-planes,  and  the  two  are 
symmetric  to  each  other  with  respect  to  the  .r^-plane.  They 
accordingly  intersect  in  this  plane  on  the  double  curve  (47). 

A  given  curve  of  the  system  cuts  the  -ty^-plane  in  three 
double  points  one  on  each  of  the  straight  lines  x  =  0,  x  = 
+  '^  Z  y '■>  the  _>/;-plane,  in  a  double  point  on  the  _>/-axis,  two 
double  points  on  the  double  curve  and  two  ordinary  points 
on  the  geodesic  ;  and  the  .t-^^-plane,  in  eight  ordinary  points, 
two  in  each  quadrant,  one  on  each  branch  of  the  double  curve. 

The  qj  Curves. — These  are  of  degree  eight.  When  q)  is  fixed, 
the  substitutions  ±  A,  ±  i/A  show  the  ciirve  to  be  symmetric  to 
the  xy-'pl&nQ  and  the  origin.     Since,  except  in  the  case  of  the 


25 


straight  lines,  the  intersections  of  the  curves  with  the  xy- 
plane  are  imaginary  or  isolated,  each  curve  consists  of  two 
non-intersecting  branches.  Curves  cp  and  -t  —  q)  are  sym- 
metric to  each  other  with  respect  to  the  xz-  and  v^-planes 
and  intersect  on  the  double  curves  in  those  planes.  Through 
each  point  on  the  >'-axis  between  ±  4/3  pass  six  (^-curves. 
The  direction  cosines  of  the  tangent  are 

[X^  -f-  i)  sin  4(p  —  A^  (A.'"^  -|-   i)  sin  2<p 


(A^-hj)  cos  4(p  +  X^  (A^  4-  I)  cos  2cp 


P-l/i 


2  A  (A"*  —  i)  sin  3^ 


where 


P^h 


V(A^-|-i)2-f  A*  (A^+i)-'^  2  A^  (A'*^  +  I)  (A'^-f  i)  cos  6<p-f-  4  A^  (A^— i)-  sin  3<p 

These  values  show  that  the  ^-curves,  with  the  exception  of 
the  geodesies,  cut  the  axis  of  z  orthogonally,  and  all  become 
parallel  to  the  .t'r-plane  at  infinity. 

23.  Model  of  the  Surface. — The  following  values  multi- 
plied throughout  by  two  furnished  the  data  for  a  skeleton 
model  of  the  surface,  made  on  the  same  plan  as  that  of  the 
double  surface. 


Geodesic  in  yz- 

Plane. 

Double  Curve  in  j^'-Plane. 

A 

y 

s 

A 

y 

z 

I 

0 

1-34 

I 

0 

1-34 

1.2 

—  •03 

1.54 

1. 14 

.02 

1.44 

1.4 

—  .17 

2.07 

1.299 

.14 

1.76 

1.6 

—  •51 

2.89 

1.48 

■49 

2.32 

1.686 

—  •75 

3^34 

1.686 

1.22 

3^19 

Double  Curve  in  xs'-Plane. 


Branch  I. 
{M  taken  with  upper  signs.) 

Branch  II. 
{M  taken  with  lower  signs.) 

A 

X 

z 

A 

X 

S 

I 
1. 14 
1.299 
1.48 
1.686 

0 

.01 

.09 

•32 
.84 

1.34 
1.44 

1.76 
2.36 

3^i8 

I 

1. 14 
1.299 
1.48 
1.686 

0 
.12 

•49 
1. 16 

2.26 

1.34 
1.40 
1.60 

1-95 
2.36 

26 


Intersections  of  Certain  A-  and  ^j-Ccjrves. 


n 

in 

1"^ 

47r 

A 

qj  =z  0 

^  =  T7 

^  =  ^7 

(P=-  — 

<P  =  — 

24 

24 

24 

24 

.1- 

0 

0 

0 

0 

0 

f°  =  i 

7 

0 

0 

0 

0 

0 

IT 

z 

0 

—  -51 

—  .94 

—1.23 

—  '•33 

e    24 

X 

0 

.07 

.10 

.09 

.01 

=  1. 14 

y 

—  .54 

—  -49 

—  -37 

—  .19 

0 

z 

0 

—  -55 

—  1.02 

—  1-33 

—  1.44 

2jr 

s  24 

X 

0 

•17 

.27 

.24 

.07 

=    1.299 

y 

—  1. 12 

—  1.07 

-  -79 

—  -39 

.04 

z 

0 

—  .68 

—  1.25 

— 1.63 

—1.77 

^ 

E    24 

X 

0 

•35 

.56 

•53 

.24 

=    1.48 

y 

— 2.02 

-1.83 

—1.32 

—  .61 

.14 

z 

0 

—  -91 

—  1.68 

— 2.19 

—2.37 

47r 

£    24 

X 

0 

.68 

1. 10 

I. II 

.65 

=  1.686 

y 

—3-24 

—2.93 

—2.08 

—  .88 

•37 

z 

0 

—1.28 

-2.36 

—3-09 

—3-34 

My  sincere  thanks  are  due  to  Professor  F.  D.  Sherman,  of 
Columbia  University,  who  from  the  wire  model  made  the 
drawing-  of  the  surface  reproduced  in  Plate  II. 


III.  Properties  of  the  Primitive  Surfaces  in  General. 

When  the  methods  used  in  the  preceding  pages  are  ap- 
plied to  the  discussion  of  the  primitive  surfaces  double  and 
conjugate  for  a  general  value  of  /^,  results  are  obtained  that 
are  presented  below  in  tabular  form.  By  reference  to  page  4 
the  results  due  to  the  paper  already  cited  may  be  dis- 
tinguished. In  treating  the  conjugate  surfaces,  only  those 
points  in  which  they  differ  from  the  corresponding  double 
surfaces  will  be  noted. 


27 


Double 


Priniittvc  Surfaces. 


IX  either  odd  or  even. 


Conjugate 


1 .  Properties  of  the  minimal  curve  : 
Order,  2/<  -f-  4;  class,  2//  -\-2;  rank, 
VI  +  4- 

Multiplicity  of  circle  at  infinity  on 
tangential  surface  is  i. 

Number  of  coincident  points  at 
infinity,  2/«  +  4. 

Plane  of  osculation  at  infinitely 
distant  points  falls  into  plane  at 
infinity,  and  tangent  line  becomes 
tangent  to  imaginary  circle. 

Stationary  points  are  furnished  by 
the  roots  of  W^t^  +  2  =  (  _  i)^  +  '• 

2.  Class  of  the  Surface,  2/<  -f-  3- 
Order  of  the  Surface,  (2//4-3)(/'  +  2). 

3.  Infinitely  distant  elements  lie 
on  three  straight  lines  of  multiplicity 
[n  +  2)  with  equations. 

^^  =  o  r  =  o 
X  -\-  i  y  =:  o  r  =  o 
X  —  i  y  ^  o     r  ^  o 

4.  All  plane  lines  of  curvature 
on  the  surface  are  geodesies. 

No  asymptotic  lines  exist  corre- 
sponding to  small  circles  on  the 
sphere. 

The  surface  possesses  (//  +  i) 
geodesies  lying  in  planes  passing 
through  the  axis  of  z.  To  these 
planes  the  .I'S'-plane  always  belongs. 

The  surface  possesses  (//  -f-  i) 
straight  lines  lying  in  the  .r_y-plane 
and  passing  through  the  origin. 

5.  There  are  no  vertical  planes  of 
symmetry,  nor  axes  of  symmetry 
that  lie  in  the  ay/-plane,  aside  from 
those  determined  by  the  geodesies 
and  straight  lines  mentioned  above. 


Class,  2(2//  -1-  3). 

Order,  2(2//  +  3)  {n  +  2). 

The  real   line    is    of    multiplicity 

2(/<  +  2). 


To  these  planes  the  .rr-plane  does 
not  always  belong. 

Among  the  {h  +  1)  lines,  the  axis 
of/  is  always  included. 

5.  There  are  (/<  +  i)  additional 
planes  and  (/<  -f-  O  ^^^s  of  symmetry, 
bisecting  the  angles  between  the 
planes  and  lines  of  the  first  set. 
The  planes  of  the  second  set  do  not 
contain  geodesies.  The  lines  do 
not  lie  on  the  surface. 

Also  the  .t;y-plane  and  3-axis  are 
respectively  a  plane  and  axis  of 
symmetry. 


/^  eve7t. 


I.  The  surface  possesses  a  geo- 
desic in  the  .ry-plane.  The  curve 
is  a  hypocycloid  with  (//  +  i)  cusps. 
It  is  cut  by  each  straight  line  of  the 
surface  in  a  cusp,  an  ordinary  point 

and double  points. 

The  .rjz-plane  is  a  plane  of  sym- 
metry containing  a  geodesic. 


I.  The  s-axis  between  the  points 
±  \li-i  4- 1  is  connected  with  2  (//  + 1 ) 
real  nappes  of  the  surface.  Outside 
of  these  limits  it  is  isolated. 


The  xy-plane  is  a  plane  of  sym- 
metry, not  containing  a  geodesic. 


28 


2.  The  s'-axis  does  not  lie  on  the 
surface,  and  is  not  an  axis  of  sym- 
metry. 

3.  The  (/<  +  0  vertical  planes  of 
symmetry  pass  through  the  (//  +  i) 
straight  lines  and  both  pass  through 
the  cusps  of  the  hypocycloid.  To 
the  lines  always  belongs  the  axis  of  x. 

4.  Pairs  of  conjugate  stationary 
points  of  the  minimal  curve  are 
symmetric  to  the  .ij-plane,  and  the 
middle  points  of  lines  joining  such 
pairs  are  the  cusps  of  the  geodesic 
hypocycloid. 

5.  The  curves  of  intersection  in 
the  .ry-plane  are 

{}i  -{-  i)  finite 
straight  lines  of 
multiplicity 

{H  +  2) (//+  i)(«  +  2) 

I  line  at  infinity 
of       multiplicity 

(/"  +  2) //  +  2 

/</2  isolated  hy- 
pocycloids  of 
multiplicity  2, 
degree  {/i  +2)..  /'  {11  +  2) 
I  geodesic  hy- 
pocycloid of 
multiplicity  i, 
degree  [n  +  2). ._ /<  +  2 

(2//  +  3)(/'  +  2) 
All  of  the   hypocycloids  may  be 
derived  from  any  one  of  the  number 
by  factors  of  proportionality. 

The  (//  +  i)  straight  lines  are 
tangent  to  the  geodesic  at  the  cusps 
and  from  these  points  outward  to 
infinity  are  each  connected  with  two 
real  nappes  of  the  surface. 

6.  The  sections  by  the  vertical 
planes  of  symmetry  are  of  two  types 
differing  in  the  position  of  corre- 
sponding curves  with  respect  to  the 
axis  of  z. 

The  curves  of  intersection  in  the 
;r2'-plane  are: 

I   straight  line  (.r-axis)  of  mul- 
tiplicity (  //  +  2). 
I  geodesic  of  multiplicity  i,and 

degree  (/<  -f  2). 
I  multiple  curve. 
The  geodesic  resembles  an  or- 
dinary parabola.  It  cuts  the  .I'-axis 
orthogonally  at  the  simple  point  on 
the  geodesic  hypocycloid.  It  is 
symmetric  to  the  jr-axis  and  cuts  the 
^■-axis  in  two  real  points.  Through 
these  pass  all  the  (/<  +  i)  vertical 
geodesies. 


2.  The  5--axis  lies  on  the  surface, 
and  is  an  axis  of  symmetry. 

3.  The  planes  of  the  geodesies 
and  the  straight  lines  of  the  surface 
are  still  united  in  position ;  but  the 
hypocycloids  are  imaginary.  To 
the  planes  always  belongs  the  yz- 
plane. 

4.  Pairs  of  conjugate  stationary 
points  are  symmetric  to  the  axis  of  z 
and  furnish  the  points  ±  4//<  -|-  i. 


5.  The  curves  of   intersection  in 
the  .ry- plane  are  : 
(/<    +    i)     finite 

straight    hues 

of  multiplicity 

2(/'  +  2) 2(/<  +  I)  (/<  +  2) 

I     Straight    line 

at    infinity   of 

multipli  city 

2{n  +  2), 2  [1.1  +  2) 

fi  +  I  imaginary 
hypocycloids 

(including  the 

origin)       of 

multiplicity  I, 

and    degree 

2{H  +2) 2(/<+  !)(/<  + 2) 

2(2A  +  3)(/'  +  2) 


The  (/<  +  1)  straight  lines  through- 
out their  course  are  each  connected 
with  two  real  nappes  of  the  surface. 


6.  The  sections  are  of  two  types 
according  as  the  cutting  plane  does 
or  does  not  contain  a  geodesic.  The 
j/s-plane  furnishes  a  section  of  the 
first  type ;  the  ^^•-plane,  of  the 
second. 

The  curves  in  the^y^'-plane  are: 

I  conical  point,  the  origin, 

I   straight  line  (the  axis  of  s)  of 

multiplicity,  2  (/<  +  1). 
I  geodesic  of  multiplicity   i,  and 

degree  2  (/<  +  2). 
I  multiple  curve. 

The  geodesic  is  symmetric  to  the 
axes.  It  has  two  cusps  on  the  axis 
of  s  at  ±  4//<  +  I,  and  cuts  the 
_y-axis  in  imaginary  or  isolated  points. 
It  consists  of  two  branches,  each 
resembling  a  semi-cubical  parabola. 
The  curves  in  the  .rjr-plane  are 
I  conical  point,  the  origin. 


29 


7-  The  A  curves  are  of  degree 
2  (//  -f-  2)  with  the  exception  of  the 
geodesic  hypocycloid  and  the  real 
line  at  infinity.  These  are  of  degree 
(/<  -f- 2).  The  curves  are  symmetric 
to  all  the  planes  and  axes  of  sym- 
metry and  cut  the  ;rK-plane  in 
(/<  +  i)  double  points,  one  on  each 
of  the  straight  lines. 


8.  The  (p  curves  are  of  degree 
(/<  -(-  2).  They  are  symmetric  to 
the  .ry-plane,  which  they  cut  orth- 
ogonally on  the  geodesic.  They 
become  parallel  to  the  .tj-plane  at 
intinity. 


I  straight  line  (the  ^--axis)  of  multi- 
plicity 2  (/«  +  !)• 
I  multiple  curve. 

7.  The  A  curves  are  of  degree 
2  (/<  -\-  2)  with  the  exception  of  the 
axis  of  z.  They  are  symmetric  to 
the  planes  of  the  geodesies  and  the 
straight  lines  of  the  surface.  Two, 
A  and  '/A,  are  symmetrically  placed 
with  reference  to  the  remaining 
planes  of  symmetry  and  intersect  on 
multiple  curves  in  these  planes. 
Both  curves  correspond  to  the  one 
curve  A  on  the  double  surface. 

8.  The  q)  curves  are  of  degree 
i[h  4-  2).  They  are  symmetric  to 
the  origin  and  the  ,tj-plane.  Each 
consists  of  two  non-intersecting 
branches  symmetrically  placed  with 
reference  to  the  .ij-plane  and 
cutting  the  axis  of  2  orthogonally 
between  ±  4/«  -f-  i.  2  (/<  -|-  i)  of 
the  curves  pass  through  each  non- 
isolated point  of  the  axis  of  z,  and 
all  become  parallel  to  the  x^-plane 
at  infinity. 


H  odd 


I .  The  axis  of  z  between  the  points 
±  4//<  -f-  I  is  connected  with  (/<  +  i) 
real  nappes  of  the  surface.  Outside 
of  these  limits  it  is  isolated. 


2.  The  vertical  planes  of  the  geo- 
desies always  include  the  xz-  and 
/^■-planes. 

The  straight  lines  in  the  .ij-plane 
bisect  the  angles  between  the  planes 
of  the  geodesies. 

3.  Pairs  of  conjugate  stationary 
points  of  the  minimal  curve  are 
symmetric  to  the  z  axis,  and  fur- 
nish the  points  ;?  =  ±  4///  4-  i.  from 
which  the  geodesies  spring. 

4.  The  curves  of  intersection  in 
the  -vy-plane  are : 

{n    4-     i)    finite 

straight    lines 

of  multiplicity 

U'+  2) [h  4-  I)  (/'  +2) 

I  straight  line  at 

infinity /<  4"  2 

u  -\-\ . 

imagmary 

hypocycloids 
of  multiplicity 
I,  and  degree 

2  (;u  4-  2) . . . .   [u  -^  i)  [u  4-  2) 
(2  n  4-  3)  (/<  4-  2) 


1.  The  surface  possesses  a  geo- 
desic hypocycloid  in  the  ^j- plane. 
This  has  2(//  4-  0  cusps.  It  is  cut 
by  each  straight  line  of  the  surface 
in  two  cusps  and  (/<  —  i)  double 
points. 

2.  The  vertical  planes  of  the  geo- 
desies always  bisect  the  angles  be- 
tween the  (/<  4"  i)  straight  lines. 

These  lines  always  include  the  x- 
and  _)/-axes. 

3.  Pairs  of  conjugate  stationary 
points  are  symmetric  to  the  .ij-plane 
and  furnish  the  cusps  of  the  geodesic 
hypocycloid. 

4.  Curves  in  the  .rj/-plane: 
(/<   4-    i)      finite 

straight  lines 
of  multiplicity 
2  (/<  4-  2)...  .    2(//4-  i)  (/<4-  2) 

I  straight  line 
at  infinity  of 
m  u  1 1  i  p  1  i  c  ity 
2  (/<  4-  2) 2(/<  4-  2) 

I  geodesic  hypo- 
cycloid of  de- 
gree 2  (/<  4-  2)  2(/<  4-  2) 

isolated 

2 

hypocycloids 

of  multiplicity 

2,       degree 

2(/<  4-2)  ....    2(/<— i)  (//4-2) 


30 


The  straight  lines  are  connected 
with  real  parts  of  the  surface 
throughout  their  course.  One  real 
nappe  passes  through  each. 

5.  The  sections  by  the  vertical 
planes  of  symmetry  are  of  two  types 
differing  in  the  position  of  corre- 
sponding curves  with  respect  to  the 
;ir-axis. 

The  curves  of  intersection  in  the 
_y5'-plane  are  : 

I  straight  line  (s'-axis)  of  multi- 
plicity {/t  +  2). 

I  geodesic  of  degree  {fi  +  2). 

I  multiple  curve. 

The  geodesic  is  symmetric  to  the 
axis  of  z  on  which  it  has  a  cusp 
at  —  4//<-)-i.  Its  intersections  with 
the  .r-axis  are  imaginary.  It  re- 
sembles a  semi-cubical  parabola. 


6.  The  A  curves  are  of  degree 
2  (/<  -|-  2)  with  the  exception  of  the 
s'-axis,  and  the  real  line  at  infinity. 
The  curves  are  symmetric  to  the 
planes  and  lines  of  symmetry  and 
cut  the  ,ry-plane  in  2  (//-(-  i)  ordi- 
nary points,  two  symmetrically 
placed  with  reference  to  the  origin 
on  each  of  the  (/<  -}-  i)  straight 
lines. 

7.  The  cp  curves  are  of  degree 
(u  -f-  2).  They  are  symmetric  to 
the  ^--axis,  and  with  the  exception  of 
the  (/<  -|-  i)  straight  lines  have  no 
real  points  in  the  ,tj-plane.  /<  +  i  of 
the  curves  pass  through  each  non- 
isolated point  of  the  ^--axis.  All 
become  parallel  to  the  aj-plane  at 
infinity. 


I  h  y  p  o  c  ycloid 
(the  origin) 
of  multiplicity 
I,       degree 

2(«-f-2) 


2{U  +  2) 


2(2//4-3)(/,  +  2) 

The  straight  lines  are  tangent  to 
the  geodesic  hypocycloid  at  the 
cusps  and  proceed  outward  from 
these  points  in  connection  with  two 
real  nappes. 

5.  The  sections  are  of  two  types 
according  as  the  planes  do  or  do  not 
contain  geodesies.  None  of  the 
planes  of  the  geodesies  coincides 
with  a  co-ordinate  plane.  But  by 
rotation  of  the  surface  one  may  be 
brought  into  coincidence  with  the 
j2-p\a.ue.  The  curves  of  intersec- 
tion are  found  to  be  : 

I  conical  point,  the  origin. 

I  geodesic  of  degree  2{u  +  2). 

I  multiple  curve. 

The  geodesic  consists  of  two 
branches,  each  of  the  same  type  as 
the  vertical  geodesies  of  the  double 
surfaces  /<  even.  These  branches 
intersect  on  the  ^•-axis. 

The  curves  of  intersection  in  the 
j'^'-plane  are : 

I  conical  point,  the  origin. 

I  straight  line  ( r-axis)  of  multi- 
plicity 2(/<  +  2). 

I  multiple  curve. 

6.  The  A  curves  are  of  degree 
2{/ii  -f-  2).  Each  is  symmetric  with 
respect  to  the  ^•-axis  and  to  the 
vertical  planes  of  the  geodesies. 
Curves  A  and  ^/A.  are  symmetric  to 
each  other  with  respect  to  the  re- 
maining planes  of  symmetry,  and 
give  by  their  intersection,  multiple 
curves  in  those  planes.  Both  corre- 
spond to  a  curve  A  on  the  double 
surface. 

7.  The  (p  curves  are  of  degree 
2{u  -f-  2).  They  are  symmetric  to 
the  origin  and  consist  of  two  non- 
intersecting  branches  symmetrically 
disposed  to  the  ^-axis.  They  cut 
the  geodesic  hypocycloid  orthogon- 
ally and  become  parallel  to  the 
-ij-plane  at  infinity. 


PLATE    I, 


FIG.  1. 


FIG.   2. 


PLATE    II. 


'  A  T  TFfl 


K/i-iKl^TJ&a-r^ay.'-',  * 


■^"■,f?  ;':;•&■; 


